Positively Homogeneous Functions Revisited

The paper deals with positively homogeneous functions defined on a finite‐dimensional space. Our attention is mainly focused on those subspaces of positively homogeneous functions that are important in nonsmooth analysis and optimization: the subspace of continuous positively homogeneous functions, of Lipschitz continuous positively homogeneous functions, of difference sublinear functions, and of piecewise linear functions. We reproduce some known results and present a number of new ones, in particular, those that concern Lipschitz continuous positively homogeneous functions.